Problem 323 - Calculus Volume 3

Question

True or False? Justify your answer with a proof or a counterexample. The equations \(x=\cosh (3 t), \quad y=2 \sinh (3 t)\) represent a hyperbola.

Step-by-Step Solution

Answer

True, the given equations represent a hyperbola.

Step 1: Recall Definitions

A hyperbola is defined by the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for hyperbolas centered at the origin with the transverse axis along the x-axis. Another form is \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) for hyperbolas with the transverse axis along the y-axis.

Step 2: Express Hyperbolic Identities

Recall the hyperbolic identity \(\cosh^2(u) - \sinh^2(u) = 1\). Given \(x = \cosh(3t)\) and \(y = 2\sinh(3t)\), we can separate these to relate to known identities.

Step 3: Manipulate Equations

Express \(\cosh(3t)\) and \(\sinh(3t)\) in terms of \(x\) and \(y\):- Let \(x = \cosh(3t)\).- Let \(\frac{y}{2} = \sinh(3t)\).Thus, from identity, \(\cosh^2(3t) - \sinh^2(3t) = 1\), we can write,\[x^2 - (\frac{y}{2})^2 = 1.\]

Step 4: Rearrange into Hyperbola Form

Multiply out the terms from Step 3:\[x^2 - \frac{y^2}{4} = 1.\]This equation is now in the form of a standard hyperbola equation \(\frac{x^2}{1} - \frac{y^2}{4} = 1\). This confirms that the equations given represent a hyperbola.

Key Concepts

Hyperbolic functions

The hyperbolic functions, similar to trigonometric functions but based on hyperbolas instead of circles, are fundamental to describing hyperbolic geometry. The primary hyperbolic functions are hyperbolic sine \( \sinh(t) \) and hyperbolic cosine \( \cosh(t) \). These functions are defined as follows:

• \( \sinh(t) = \frac{e^t - e^{-t}}{2} \)
• \( \cosh(t) = \frac{e^t + e^{-t}}{2} \)

A key identity involving these functions is \( \cosh^2(t) - \sinh^2(t) = 1 \), which mirrors the Pythagorean identity in trigonometry.
This identity is crucial in transforming problems involving hyperbolic functions into recognizable forms, such as conic sections like hyperbolas. Hyperbolic functions arise naturally in a variety of contexts, including in the description of the shape of a hanging cable, known as a catenary.

Parametric equations

Parametric equations provide a way of defining a set of related quantities as functions of an independent variable, often denoted as \( t \). In the context of conic sections, parametric equations can elegantly describe the coordinates of points lying on the curve.
For example, in the exercise provided, the parametric equations \( x=\cosh(3t), \, y=2\sinh(3t) \) define a hyperbola, where \( t \) is the parameter.

• The equation for \( x \) describes how the x-coordinate moves in relation to the hyperbolic cosine function.
• The equation for \( y \) similarly moves according to the scaled hyperbolic sine function.

To verify that these equations represent a hyperbola, we rely on the relationship between hyperbolic sine and cosine, specifically the identity \( \cosh^2(3t) - \sinh^2(3t) = 1 \). Transforming the parametric form into the standard form of a hyperbola helps validate their association.

Conic sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Depending on the angle of intersection, the resulting curves can be circles, ellipses, parabolas, or hyperbolas. Hyperbolas have distinctive characteristics and can be represented by the standard equation:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]In the exercise, the parametric forms \( x=\cosh(3t) \) and \( y=2\sinh(3t) \) were transformed using hyperbolic identities.
This process showed consistency with the hyperbola's standard form, \( \frac{x^2}{1} - \frac{y^2}{4} = 1 \). This confirms the description of a hyperbola centered at the origin with vertices along the x-axis.
Conic sections are a foundation of classical geometry and have extensive applications in physics, engineering, and astronomy, showcasing their practicality and relevance beyond theoretical math.